\section{Driver model}
There are two main types of simulation models for traffic systems: microscopic
and macroscopic. A microscopic model describe each vehicle individually while macroscopic
models describe the complete traffic flow. Since microscopic models describe the position
and velocity of each car in the simulation, they can easier be compared with
empirical data from real experiments than macroscopic models. The Intelligent Driver Model (IDM)
is a car-following model and belongs to the deterministic kind of microscopic
models \cite{idm}.

The IDM controls the position of the car on a single-lane road. The position depends on the velocity and acceleration of the car. Acceleration is described by the velocity \begin{math}v_\alpha\end{math} and distance to the car in front \begin{math}s_\alpha\end{math}. These two parts are related to the desired velocity \begin{math}v_0\end{math} and effective desired distance \begin{math}s^\ast\end{math}. The equation for acceleration then becomes:
\begin{equation}\label{driver_acc}\dot{v_\alpha} = a\left
(1-(\frac{v_\alpha}{v_0})^\delta-(\frac{s^\ast}{s_0})^2 \right)\end{equation}
Desired distance between the cars is calculated from minimum distance \begin{math}s_0\end{math}, time headway \begin{math}T\end{math} and difference in velocity \begin{math}\Delta v = v_\alpha - v_{\alpha + 1}\end{math}.
\begin{equation}\label{desireddist}s^\ast = s_0 + max \left (v_\alpha T +
\frac{v_\alpha \Delta v}{2\sqrt{ab}}\right )\end{equation}
